Recently, Sean wrote a blog post about the 'sleeping beauty problem' and came out in favor of the 1/3 solution.
Naturally, Lubos had to respond with an obvious 1/2 answer and finally Joe Polchinski jumped in defending 1/3.
I wrote about this puzzle nine years ago on that other blog and people are still discussing it, because it has all the ingredients of a nice paradoxon: i) it is easy to calculate the probabilities and ii) there is an interesting ambiguity on how to use "probability" in this case, with the same outcome unknowingly sampled twice.
It seems that Sean honestly thinks that this ambiguity can be resolved using the many worlds interpretation, but I would point out that it actually supports Bohr's philosophy of complementarity: In general we cannot assign probabilities to a local object (in this case the coin), but we have to understand them in the context of an observational situation. In other words, the 'sleeping beauty problem' supports the Copenhagen interpretation.
8-)
added later: After re-reading my old post about this, I actually like the 3/8 solution.
Btw one way to undermine the 1/3 solution based on bet size (a la Polchinski) is to point out that the same bet size optimizes the outcome if SB does not fall asleep the 2nd time (but still has to bet twice the same amount). In this case the probability surely is not 1/3.
added even later: Joe posted yet another comment "which may make it clearer" why the correct answer is 1/3, while Lubos posted a whole new blog post defending the "obvious" 1/2 solution with an interesting modification of the original puzzle: Instead of just being put to sleep twice, with small probability 1/N 'sleeping beauty' is kidnapped by the CIA and tortured - waking up and put to sleep again and again N times; as N goes to infinity, the probability of being kidnapped by the CIA should go to zero.
added much later: Finally a commenter says what needs to be said about all this. It took long enough 8-)